LBA: Linear Bifurcation Analysis (Linear Buckilng)

Sometimes you just want to have a quick and rough estimate of model buckling capacity. Linear Bifurcation Analysis (also called “Buckling” in many software packages) does exactly this! In the analysis solver gives you the minimal multiplier (sometimes denoted as α, or called eigenvalue) such that when you multiply the applied loads with this multiplier you will obtain loads that are critical for analysed model. Critical loads cause model to be unstable (I.e. due to buckling) when such load is applied. Also you receive a geometry of this stability failure (I will call it “shape”, but this is part of the eigenvalue solution).

Lets use an example to illustrate this better: Imagine we have a column (pinned at both ends with vertical release on top) that is compressed. Critical load for such column is easy to calculate for the first eigenvalue shape according to Euler’s Formula. We can however make an LBA calculations instead of using said formula to receive that load:

Applied force was 1kN, which is definitely too small to cause instability failure in 5m long rectangular pipe (used in the example). This is however not important in LBA since the multiplier (eigenvalue) is 337.9 – which means that the critical load causing buckling shown in eigenvalue 1 (on the left) is 1kN * 337.9 = 337.9kN. If 1000kN would have been applied eigenvalue would be 0.3379 and the result would be the same.

Also please note that more than one critical load can be calculated – here 3 first eigenvalues were plotted (pipe is supported in second plane). Theoretically second and third shape cannot be obtained in reality (since the column will fail due to instability at much smaller load given for eigenvalue 1 on the left), but sometimes knowing higher eigenvalues and corresponding eigenvalue shapes can be of use.

One last thing that should be explained is that eigenvalue shape is not a deformed shape of the model… it’s rather a “way” instability will be reached. Difference is, that usually deformations are without units and maximal value is simply 1 in this analysis. This is because after instability deformations rise to infinity (this is why the model is unstable) and as such, there is no way to actually give “real value” to those deformations. Plotting those deformations however make sense, as it is important to know which part of the model move the most, and which stay in place (in other words what part of the model is unstable and how) – this is why deformations are normalized to 1.

LBA analysis usages:

Eigenvalues shows which part of the model is the weakest due to stability – this is helpful in first iterations of optimisation process.

LBA can be used in model verification: if something got disconnected by accident it will usually became unstable, and the eigenvalue shape shows which parts move in relation to other parts (so where the connection failed).

Imperfection forms (shape of the deformed, imperfect model) for more complex analysis can be made based on eigenvalue shapes.

Critical load obtained from this analysis can be used in many code-related procedures of dimensioning structural elements. In civil engineering many code-related procedures are based on slenderness (calculated with usage of critical forces / moments that can be obtained in LBA). Usually codes provide simplified rules for calculation of critical forces / moments, but for more complex cases usage of LBA is very useful.

At the end of this post I wish to address important issue: There is a certain misconception with LBA. This analysis does not estimate capacity due to instability (even threw it is often called “Buckling” that may be misleading). This is only a supporting analysis in my opinion that helps you to better understand how the model works. This is because outcomes from LBA are geometrically linear, and all instability problems are geometrically nonlinear in nature, and as such LBA is not fitted for solving said problems. This is true for all models and can be easily proven: there is a difference between Euler’s Force for column from our example above, and real compressive capacity of this column (such capacity can be even 2 times smaller than Euler’s force). For linear elements there are usually code regulations for capacity (as as such one can easily find a mistake), but in more complex example of shells this is not as obvious, and capacity estimated with LBA can be even several times higher that real-life capacity of the structure! This effect cannot be omitted and LBA should never be the “main” tool for design.

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